Summary Brief periods of neural activity trigger a vascular response with a complex but stable form. This hemodynamic response function (HRF) is extensively exploited in popular imaging methods such as functional magnetic resonance imaging (fMRI) and reflectance based optical imaging. Proper interpretation and utilization of these imaging techniques requires a detailed understanding the HRF. Moreover, because the brain is continually active, transient responses are likely to be important in normal function of the healthy brain. Better understanding of the HRF will enable research efforts that rely on resting-state or default-mode activity that is clearly evident in the brain. Finally, cerebrovascular pathology is likely to affect such transient responses, so a more complete understanding of their normal character should enable their investigation as diagnostics of incipient or extant pathology. The physiology and physics of the HRF are not well understood. A standard model has been developed that incorporates a non-linear compliant element (balloon) to explain flow and volume changes. However, some specific predictions of the balloon model have not been confirmed experimentally, and there is now great controversy on the subject. We have developed a new model that combines a linear flow description with one-dimensional convection-diffusion oxygen transport. The flow model includes the inertial dynamics of moving blood, which should not be neglected in the larger arterioles that mechanically mediate the flow response. Our full model has provided remarkably accurate fits to tissue oxygen measurements performed with polarographic probes consequent to brief (few second) neural stimulation events. This model also provides an explanation for the neurovascular coupling utilized in resting-state fMRI. We propose further development and testing of our model. Specifically, we want to improve the quality of its intravascular oxygen concentration predictions, and to enable examination of oxygen mass-balance issues. Aim 1 is to expand the model in several ways: by adding the dynamics of oxygen dissociation from hemoglobin, and by adding terms to deal with alternative oxygen transport mechanisms such as oxygen consumption by endothelial tissue. Aim 2 is to further test our model by fitting additional tissue oxygen measurements obtained using polarographic probes. Aim 3 is to use two-photon optical imaging to measure flow speeds in small arterioles and capillaries, and how they change consequent to brief neural stimulation. We will test how well these measurements are fit by our linear flow model as well as previous non-linear models. The proposed work will further vet our model, bringing it into the mainstream as a simple but powerful analytic description for cerebrovascular hemodynamics.